# std::cauchy_distribution

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< cpp‎ | numeric‎ | random

Defined in header `<random>` |  |
template< class RealType = double > class cauchy_distribution; |  | (since C++11)

Produces random numbers according to a Cauchy distribution (also called Lorentz distribution):

  \({\small f(x;a,b)={(b\pi{[1+{(\frac{x-a}{b})}^{2}]} })}^{-1}\)f(x; a,b) = ⎛
  ⎜
  ⎝bπ ⎡
  ⎢
  ⎣1 + ⎛
  ⎜
  ⎝

  x - a
  b

  ⎞
  ⎟
  ⎠2
  ⎤
  ⎥
  ⎦⎞
  ⎟
  ⎠-1

`std::cauchy_distribution` satisfies all requirements of RandomNumberDistribution.

### Template parameters

RealType | - | The result type generated by the generator. The effect is undefined if this is not one of float, double, or long double.

### Member types

Member type | Definition
`result_type` (C++11) | RealType
`param_type` (C++11) | the type of the parameter set, see RandomNumberDistribution.

### Member functions

(constructor) (C++11) | constructs new distribution (public member function)
reset (C++11) | resets the internal state of the distribution (public member function)
##### Generation
operator() (C++11) | generates the next random number in the distribution (public member function)
##### Characteristics
ab (C++11) | returns the distribution parameters (public member function)
param (C++11) | gets or sets the distribution parameter object (public member function)
min (C++11) | returns the minimum potentially generated value (public member function)
max (C++11) | returns the maximum potentially generated value (public member function)

### Non-member functions

operator==operator!= (C++11)(C++11)(removed in C++20) | compares two distribution objects (function)
operator<<operator>> (C++11) | performs stream input and output on pseudo-random number distribution (function template)

### Example

Run this code

    #include <algorithm>
    #include <cmath>
    #include <iomanip>
    #include <iostream>
    #include <map>
    #include <random>
    #include <vector>

    template<int Height = 5, int BarWidth = 1, int Padding = 1, int Offset = 0, class Seq>
    void draw_vbars(Seq&& s, const bool DrawMinMax = true)
    {
        static_assert(0 < Height and 0 < BarWidth and 0 <= Padding and 0 <= Offset);

        auto cout_n = [](auto&& v, int n = 1)
        {
            while (n-- > 0)
                std::cout << v;
        };

        const auto [min, max] = std::minmax_element(std::cbegin(s), std::cend(s));

        std::vector<std::div_t> qr;
        for (typedef decltype(*std::cbegin(s)) V; V e : s)
            qr.push_back(std::div(std::lerp(V(0), 8 * Height,
                                            (e - *min) / (*max - *min)), 8));

        for (auto h{Height}; h-- > 0; cout_n('\n'))
        {
            cout_n(' ', Offset);

            for (auto dv : qr)
            {
                const auto q{dv.quot}, r{dv.rem};
                unsigned char d[]{0xe2, 0x96, 0x88, 0}; // Full Block: '█'
                q < h ? d[0] = ' ', d[1] = 0 : q == h ? d[2] -= (7 - r) : 0;
                cout_n(d, BarWidth), cout_n(' ', Padding);
            }

            if (DrawMinMax && Height > 1)
                Height - 1 == h ? std::cout << "┬ " << *max:
                              h ? std::cout << "│ "
                                : std::cout << "┴ " << *min;
        }
    }

    int main()
    {
        std::random_device rd{};
        std::mt19937 gen{rd()};

        auto cauchy = [&gen](const float x0, const float 𝛾)
        {
            std::cauchy_distribution<float> d{x0 /* a */, 𝛾 /* b */};

            const int norm = 1'00'00;
            const float cutoff = 0.005f;

            std::map<int, int> hist{};
            for (int n = 0; n != norm; ++n)
                ++hist[std::round(d(gen))];

            std::vector<float> bars;
            std::vector<int> indices;
            for (auto const& [n, p] : hist)
                if (float x = p * (1.0 / norm); cutoff < x)
                {
                    bars.push_back(x);
                    indices.push_back(n);
                }

            std::cout << "x₀ = " << x0 << ", 𝛾 = " << 𝛾 << ":\n";
            draw_vbars<4,3>(bars);
            for (int n : indices)
                std::cout << std::setw(2) << n << "  ";
            std::cout << "\n\n";
        };

        cauchy(/* x₀ = */ -2.0f, /* 𝛾 = */ 0.50f);
        cauchy(/* x₀ = */ +0.0f, /* 𝛾 = */ 1.25f);
    }

Possible output:

    x₀ = -2, 𝛾 = 0.5:
                        ███                     ┬ 0.5006
                        ███                     │
                    ▂▂▂ ███ ▁▁▁                 │
    ▁▁▁ ▁▁▁ ▁▁▁ ▃▃▃ ███ ███ ███ ▂▂▂ ▁▁▁ ▁▁▁ ▁▁▁ ┴ 0.0076
    -7  -6  -5  -4  -3  -2  -1   0   1   2   3

    x₀ = 0, 𝛾 = 1.25:
                                    ███                                 ┬ 0.2539
                                ▅▅▅ ███ ▃▃▃                             │
                            ▁▁▁ ███ ███ ███ ▁▁▁                         │
    ▁▁▁ ▁▁▁ ▁▁▁ ▁▁▁ ▃▃▃ ▅▅▅ ███ ███ ███ ███ ███ ▅▅▅ ▃▃▃ ▂▂▂ ▁▁▁ ▁▁▁ ▁▁▁ ┴ 0.0058
    -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   9

### External links

Weisstein, Eric W. "Cauchy Distribution." From MathWorld — A Wolfram Web Resource.

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